There is an outlet sold at The Home Depot which will allow you cut your own hole in gutter and insert outlet to connect downspout to. It comes in 2x3 and 3x4 depending on the size of pipe you want to use.
This product is for use with 5 inch gutters. In order to determine if your gutters measure 5 inches, use a measuring tape or ruler and check the distance the gutter protrudes from your roof.
Yes, your description is exactly what this outlet is for. You will need sealant to make sure it doesn't leak.